Answer :
To find the solution to the equation \(\frac{2}{3}x + \frac{8}{3} = 2^x\), we will analyze and attempt to solve it step-by-step.
### Step 1: Simplify the Equation
Firstly, let’s write the equation in a more manageable form:
[tex]\[ \frac{2}{3}x + \frac{8}{3} = 2^x \][/tex]
### Step 2: Test the Possible Integer Solutions
Given the multiple-choice answers \(x = -1\), \(x = -2\), \(x = 2\), and \(x = 1\), we will substitute these values into the equation to check if any of them satisfy the equation.
#### Testing \(x = -1\):
[tex]\[ \frac{2}{3}(-1) + \frac{8}{3} = 2^{-1} \][/tex]
[tex]\[ -\frac{2}{3} + \frac{8}{3} = \frac{1}{2} \][/tex]
[tex]\[ \frac{6}{3} = \frac{1}{2} \][/tex]
[tex]\[ 2 \neq \frac{1}{2} \][/tex]
Thus, \(x = -1\) is not a solution.
#### Testing \(x = -2\):
[tex]\[ \frac{2}{3}(-2) + \frac{8}{3} = 2^{-2} \][/tex]
[tex]\[ -\frac{4}{3} + \frac{8}{3} = \frac{1}{4} \][/tex]
[tex]\[ \frac{4}{3} = \frac{1}{4} \][/tex]
[tex]\[ 1.33 \neq 0.25 \][/tex]
Thus, \(x = -2\) is not a solution.
#### Testing \(x = 2\):
[tex]\[ \frac{2}{3}(2) + \frac{8}{3} = 2^2 \][/tex]
[tex]\[ \frac{4}{3} + \frac{8}{3} = 4 \][/tex]
[tex]\[ \frac{12}{3} = 4 \][/tex]
[tex]\[ 4 = 4 \][/tex]
Thus, \(x = 2\) is a solution.
#### Testing \(x = 1\):
[tex]\[ \frac{2}{3}(1) + \frac{8}{3} = 2^1 \][/tex]
[tex]\[ \frac{2}{3} + \frac{8}{3} = 2 \][/tex]
[tex]\[ \frac{10}{3} = 2 \][/tex]
[tex]\[ 3.33 \neq 2 \][/tex]
Thus, \(x = 1\) is not a solution.
### Conclusion
After testing all the provided options, the correct solution to the equation \(\frac{2}{3}x + \frac{8}{3} = 2^x\) is \(x = 2\). This matches the possible answer provided in the question options.
Thus, the solution to the equation is:
[tex]\[ x = 2 \][/tex]
### Step 1: Simplify the Equation
Firstly, let’s write the equation in a more manageable form:
[tex]\[ \frac{2}{3}x + \frac{8}{3} = 2^x \][/tex]
### Step 2: Test the Possible Integer Solutions
Given the multiple-choice answers \(x = -1\), \(x = -2\), \(x = 2\), and \(x = 1\), we will substitute these values into the equation to check if any of them satisfy the equation.
#### Testing \(x = -1\):
[tex]\[ \frac{2}{3}(-1) + \frac{8}{3} = 2^{-1} \][/tex]
[tex]\[ -\frac{2}{3} + \frac{8}{3} = \frac{1}{2} \][/tex]
[tex]\[ \frac{6}{3} = \frac{1}{2} \][/tex]
[tex]\[ 2 \neq \frac{1}{2} \][/tex]
Thus, \(x = -1\) is not a solution.
#### Testing \(x = -2\):
[tex]\[ \frac{2}{3}(-2) + \frac{8}{3} = 2^{-2} \][/tex]
[tex]\[ -\frac{4}{3} + \frac{8}{3} = \frac{1}{4} \][/tex]
[tex]\[ \frac{4}{3} = \frac{1}{4} \][/tex]
[tex]\[ 1.33 \neq 0.25 \][/tex]
Thus, \(x = -2\) is not a solution.
#### Testing \(x = 2\):
[tex]\[ \frac{2}{3}(2) + \frac{8}{3} = 2^2 \][/tex]
[tex]\[ \frac{4}{3} + \frac{8}{3} = 4 \][/tex]
[tex]\[ \frac{12}{3} = 4 \][/tex]
[tex]\[ 4 = 4 \][/tex]
Thus, \(x = 2\) is a solution.
#### Testing \(x = 1\):
[tex]\[ \frac{2}{3}(1) + \frac{8}{3} = 2^1 \][/tex]
[tex]\[ \frac{2}{3} + \frac{8}{3} = 2 \][/tex]
[tex]\[ \frac{10}{3} = 2 \][/tex]
[tex]\[ 3.33 \neq 2 \][/tex]
Thus, \(x = 1\) is not a solution.
### Conclusion
After testing all the provided options, the correct solution to the equation \(\frac{2}{3}x + \frac{8}{3} = 2^x\) is \(x = 2\). This matches the possible answer provided in the question options.
Thus, the solution to the equation is:
[tex]\[ x = 2 \][/tex]
